Theorem spcimdv 2833

Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
spcimdv.1	|- ( ph -> A e. B )
spcimdv.2	|- ( ( ph /\ x = A ) -> ( ps -> ch ) )

Assertion

Ref	Expression
spcimdv	|- ( ph -> ( A. x ps -> ch ) )

Distinct variable groups:   x, A   ph, x   ch, x

Allowed substitution hints:   ps( x)   B( x)

Proof of Theorem spcimdv

Step	Hyp	Ref	Expression
1		spcimdv.2	. . . 4 |- ( ( ph /\ x = A ) -> ( ps -> ch ) )
2	1	ex 115	. . 3 |- ( ph -> ( x = A -> ( ps -> ch ) ) )
3	2	alrimiv 1884	. 2 |- ( ph -> A. x ( x = A -> ( ps -> ch ) ) )
4		spcimdv.1	. 2 |- ( ph -> A e. B )
5		nfv 1538	. . 3 |- F/ x ch
6		nfcv 2329	. . 3 |- F/_ x A
7	5, 6	spcimgft 2825	. 2 |- ( A. x ( x = A -> ( ps -> ch ) ) -> ( A e. B -> ( A. x ps -> ch ) ) )
8	3, 4, 7	sylc 62	1 |- ( ph -> ( A. x ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1361   = wceq 1363   e. wcel 2158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751

This theorem is referenced by:  spcdv  2834  rspcimdv  2854